By G. George Yin, Qing Zhang

ISBN-10: 1461443458

ISBN-13: 9781461443452

ISBN-10: 1461443466

ISBN-13: 9781461443469

Prologue and Preliminaries: advent and evaluation- Mathematical preliminaries.- Markovian models.- Two-Time-Scale Markov Chains: Asymptotic Expansions of recommendations for ahead Equations.- career Measures: Asymptotic houses and Ramification.- Asymptotic Expansions of recommendations for Backward Equations.- Applications:MDPs, Near-optimal Controls, Numerical equipment, and LQG with Switching: Markov choice Problems.- Stochastic regulate of Dynamical Systems.- Numerical tools for keep watch over and Optimization.- Hybrid LQG Problems.- References.- Index

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**Extra resources for Continuous-time Markov chains and applications : a two-time-scale approach**

**Sample text**

Using the Markovian framework, the interarrival and service times follow exponential distributions with arrival and service rates λ and μ, respectively. Suppose that λ > 0 and μ > 0. Then the stationary distribution νi , i ≥ 0 of the queueing process can be easily calculated. In fact, νi = (1 − (λ/μ))(λ/μ)i provided λ < μ. For more detailed account on this and related issues as well as other examples, we refer the reader to Karlin and Taylor [105], Sharma [194], and Taylor and Karlin [204] among others.

Then (a) The process α(·) constructed above is a Markov chain. 4) 0 is a martingale for any uniformly bounded function f (·) on M. Thus Q(t) is indeed the generator of α(·). 5) P (s, s) = I, where I is the identity matrix. (d) Assume further that Q(t) is continuous in t. 6) P (t, t) = I. 6. In (c) and (d) above, the derivatives can also be written as partial derivatives, (∂/∂t)P (t, s) and (∂/∂s)P (t, s), respectively. 6) only appear in the formulas as parameters. That is, they main represents the initial and terminal conditions.

Motivated by the many applications we are interested in, a generator is deﬁned for a matrix satisfying the q-Property. Diﬀerent deﬁnitions, including other classes of matrices, may be devised as in Chung [31]. To proceed, we give an equivalent condition for a ﬁnite-state Markov chain generated by Q(t). 4. Let M = {1, . . , m}. Then α(t) ∈ M, t ≥ 0, is a Markov chain generated by Q(t) iﬀ I{α(t)=1} , . . , I{α(t)=m} − t 0 I{α(ς)=1} , . . 3) is a martingale. 2) deﬁnes a martingale. For any k ∈ M, choose fk (α) = I{α=k} .

### Continuous-time Markov chains and applications : a two-time-scale approach by G. George Yin, Qing Zhang

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